Class groups of real cyclotomic fields

Cyclotomic Units and Class Groups in Z_p extensions of real abelian fields

For a real abelian field and for an odd prime p splitting in the field, we study a map between the p-parts of the class group and of the quotient of units modulo Cyclotomic Units, respectively, along the cyclotomic Z_p-extension of the field. We determine the kernel and the cokernel of this map assuming Greenberg's conjecture on the vanishing of the lambda-invariant of the extension.

分圆域由全实域的分圆 Hecke L 值生成，II

Cyclotomic fields are generated by cyclotomic Hecke {\it L}-values of totally real fields, II

本文研究全实域上单个临界分圆 Hecke L 值能否生成相应分圆 Hecke 域，将 Jun-Lee-Sun 的平凡驯顺特征标结果推广到可解全实域上的非平凡 Hecke 特征标。

Ideal class groups of cyclotomic number fields III

Using an idea going back to Scholz, we construct unramified abelian extensions of cyclotomic extensions of number fields.

Modularity of elliptic curves over cyclotomic $\mathbb{Z}_p$-extensions of real quadratic fields

We prove that all elliptic curves defined over the cyclotomic $\mathbb{Z}_p$-extension of a real quadratic field are modular under the assumption that the algebraic part of the central value of a twisted $L$-function is a $p$-adic unit. Our result is a generalization of a result of X. Zhang, which is a real quadratic analogue of a result of Thorne.

Number of Real Critical Points of Cyclotomic Polynomials

We study the number of real critical points of a cyclotomic polynomial $Φ_{n}(x)$, that is, the real roots of $Φ_{n}^{\prime}(x)$. As usual, one can, without losing generality, restrict $n$ to be the product of distinct odd primes, say $p_{1}<\cdots<p_{k}$. We show that if the primes are "sufficiently separated" then there are exactly $2^{k}-1$ real roots of $Φ_{n}^{\prime}(x)$ and each of them is simple.

Ray Class Groups of Quadratic and Cyclotomic Fields

This paper studies Galois extensions over real quadratic number fields or cyclotomic number fields ramified only at one prime. In both cases, the ray class groups are computed, and they give restrictions on the finite groups that can occur as such Galois groups. Also the explicit structure of ray class groups of regular cyclotomic number field is given.

Greenberg's conjecture for real quadratic fields and the cyclotomic $\mathbb{Z}_2$-extensions

Let $\mathcal{A}_n$ be the $2$-part of the ideal class group of the $n$-th layer of the cyclotomic $\mathbb{Z}_2$-extension of a real quadratic number field $F$. The cardinality of $\mathcal{A}_n$ is related to the index of cyclotomic units in the full group of units. We present a method to study the latter index. As an application we show that the sequence of the $\mathcal{A}_n$'s stabilizes for the real fields $F=\mathbb{Q}(\sqrt{f})$ for any integer $0<f<10000$. Equivalently Greenberg's conjecture holds for those fields.

On the class numbers of real cyclotomic fields of conductor $pq$

The class numbers $h^{+}$ of the real cyclotomic fields are very hard to compute. Methods based on discriminant bounds become useless as the conductor of the field grows and methods employing Leopoldt's decomposition of the class number become hard to use when the field extension is not cyclic of prime power. This is why other methods have been developed which approach the problem from different angles. In this paper we extend one of these methods that was designed for real cyclotomic fields of prime conductor, and we make it applicable to real cyclotomic fields of conductor equal to the produ…

实分圆域的Kitaoka猜想

Kitaoka's conjecture for real cyclotomic fields

该研究证明了所有实分圆域上的Kitaoka猜想，明确这类域存在通用经典三元二次型当且仅当它是实二次域。

关于分圆域类数的正部分

On the plus parts of the class numbers of cyclotomic fields

本文构造了新的分圆域族，证明其类数正部分非平凡，并证明了另一无穷族类数正部分可被3整除，附数值例证。

$2$-class groups of cyclotomic towers of imaginary biquadratic fields and applications

Let $d$ be a positive square-free integer. In this paper we shall investigate the structure of the $2$-class group of the cyclotomic $\mathbb{Z}_2$-extension of the imaginary biquadratic number field $\mathbb{Q}(\sqrt{d},\sqrt{-1})$. Furthermore, we deduce the structure of the $2$-class group of cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}(\sqrt{-d})$.

A Note on the equivalence of the parity of class numbers and the signature ranks of units in cyclotomic fields

We collect some statements regarding equivalence of the parities of various class numbers and signature ranks of units in prime power cyclotomic fields. We correct some misstatements in the literature regarding these parities by providing an example of a prime cyclotomic field where the signature rank of the units and the signature rank of the circular units are not equal.

An analogue of Kummer's relation between the ideal class number and the unit index of cyclotomic fields

In this paper, we obtain a formula for the special value of Euler-Dirichlet $L$-function $L_E(s,χ)$ at $s=1$. This leads to another class number formula of $\mathbb{Q}(μ_{m})^{+}$, the maximal real subfield of $m$th cyclotomic field. From this formula, we construct a new type of cyclotomic units in $\mathbb{Q}(μ_{p^{n}})$, which implies a similar Kummer's relation between the ideal class number of $\mathbb{Q}(μ_{p^{n}})^{+}$ and the unit index.

The 3-part of the Ideal Class Group of a certain family of real cyclotomic fields

In this paper we study the structure of the $3-$part of the ideal class group of a certain family of real cyclotomic fields with $3-$class number exactly $9$ and conductor equal to the product of two distinct odd primes. We employ known results from Class Field Theory as well as theoretical and numerical results on real cyclic sextic fields, and we show that the $3-$part of the ideal class group of such cyclotomic fields must be cyclic. We present four examples of fields that fall into our category, namely the fields of conductor $3 \cdot 331$, $7 \cdot 67$, $3 \cdot 643$ and $7 \cdot 257$, an…

Class numbers in cyclotomic Z_p-extensions

For any prime p and any positive integer n, let B(p,n) denote the n-th layer of the cyclotomic Z_p-extension over the rationals. Based on the heuristics of Cohen and Lenstra, and refined by new results on class numbers of particular B(p,n), we provide evidence for the following conjecture: For all primes p and positive integers n, the ideal class group of B(p,n) is trivial.

Ideal class groups of cyclotomic number fields I

Following Hasse's example, various authors have been deriving divisibility properties of minus class numbers of cyclotomic fields by carefully examining the analytic class number formula. In this paper we will show how to generalize these results to CM-fields by using class field theory. Although we will only need some special cases, we have also decided to include a few results on Hasse's unit index for CM-fields as well, because it seems that our proofs are more direct than those given by Hasse.

A classical approach on cyclotomic fields and Fermat-Wiles theorem

This paper is submitted to Algebraic-Number-Theory Archives for validation by Number Theorists Community. It is an update of the previous versions ANT-0155, ANT-0170, ANT-0205, ANT-0237, ANT-0321, ANT-0333, and ANT-0356, of which the first four were titled `A generalization of Eichler criterium for Fermat's Last Theorem' and the last three were titled `A classical approach on Fermat-Wiles theorem'. This version contains a complete reorganization of the paper with a first part dealing with cyclotomic fields (independently of FLT) from page 10 to 72 and a second part dealing with FLT from page 7…

Real cyclotomic fields of prime conductor and their class numbers

Surprisingly, the class numbers of cyclotomic fields have only been determined for fields of small conductor, e.g. for prime conductors up to 67, due to the problem of finding the "plus part," i.e. the class number of the maximal real subfield. Our results have improved the situation. We prove that the plus part of the class number is 1 for prime conductors between 71 and 151. Also, under the assumption of the generalized Riemann hypothesis, we determine the class number for prime conductors between 167 and 241. This technique generalizes to any totally real field of moderately large discrimin…

关于某些分圆域的类数

On the class number of certain cyclotomic fields

本文构造四类无限分圆域族，证明其类数及最大实子域类数均为3的倍数，并给出计算验证。

Irreducible representations and Artin L-functions of quasi-cyclotomic fields

We determine all irreducible representations of primary quasi-cyclotomic fields in this paper. The methods can be applied to determine the irreducible representations of any quasi-cyclotomic field. We also compute the Artin L-functions for a class of quasi-cyclotomic fields.

On cyclotomic norms and the conjectures of Leopoldt and Gross-Kuz'min

We use $\ell$-adic class field theory to take a new view on cyclotomic norms and Leopoldt or Gross generalized conjectures. By the way we recall and complete some classical results. We illustrate the logarithmic approach by various numerical examples and counter-examples obtained with PARI.

On L-functions of cyclotomic function fields

We study two criterions of cyclicity for divisor class groups of functions fields, the first one involves Artin L-functions and the second one involves "affine" class groups. We show that, in general, these two criterions are not linked.

Some questions on the class group of cyclotomic fields

This article deals with a study of the structure of the class group of the cyclotomic field $K=\Q(ζ_p)$ for $p$ an odd prime number, starting from Stickelberger relation. The present state of this work leads me to set a question for all the prime numbers $h\not=p$ which divide the relative class number $h^-(K)$.

Fitting ideals of Class groups in Carlitz-Hayes cyclotomic extensions

We generalize some results of Greither and Popescu to a geometric Galois cover $X\rightarrow Y$ which appears naturally for example in extensions generated by $\mathfrak{p}^n$-torsion points of a rank 1 normalized Drinfeld module (i.e. in subextensions of Carlitz-Hayes cyclotomic extensions of global fields of positive characteristic). We obtain a description of the Fitting ideal of class groups (or of their dual) via a formula involving Stickelberger elements and providing a link (similar to the one in \cite{ABBL}) with Goss $ζ$-function.

Greenberg's conjecture and cyclotomic towers

We describe Greenberg's pseudo-null conjecture, and prove a result describing conditions under which the pseudo-null conjecture for a number field $K$ implies the conjecture for finite extensions of $K$. We then apply the result to the cyclotomic $\mathbb{Z}_p$-tower above a cyclotomic field of prime roots of unity, verifying the conjecture for a large class of cyclotomic fields.

On the Applications of Cyclotomic Fields in Introductory Number Theory

In this essay, we see how prime cyclotomic fields (cyclotomic fields obtained by adjoining a primitive p-th root of unity to Q, where p is an odd prime) can lead to elegant proofs of number theoretical concepts. We namely develop the notion of primary units in a cyclotomic field, demonstrate their equivalence to real units in this case, and show how this leads to a proof of a special case of Fermat's Last Theorem. We finally modernize Dirichlet's solution to Pell's Equation.

The first factor of the class number of the $p$-th cyclotomic field

Kummer's conjecture states that the relative class number of the $p$-th cyclotomic field follows a strict asymptotic law. Granville has shown it unlikely to be true -- it cannot be true if we assume the truth of two other widely believed conjectures. We establish a new bound for the error term in Kummer's conjecture, and more precisely we prove that $\log(h_p^-)=\frac{p+3}{4}\log(p) + \frac{p}{2}\log(2π)+\log(1-β)+O(\log_2 p)$, where $β$ is a possible Siegel zero of an $L(s,χ)$, $χ$ odd.

A note on inverses of cyclotomic mapping permutation polynomials over finite fields

In this note, we give a shorter proof of the result of Zheng, Yu, and Pei on the explicit formula of inverses of generalized cyclotomic permutation polynomials over finite fields. Moreover, we characterize all these cyclotomic permutation polynomials that are involutions. Our results provide a fast algorithm (only modular operations are involved) to generate many classes of generalized cyclotomic permutation polynomials, their inverses, and involutions.

Cohomology groups of Fermat curves via ray class fields of cyclotomic fields

The absolute Galois group of the cyclotomic field $K={\mathbb Q}(ζ_p)$ acts on the étale homology of the Fermat curve $X$ of exponent $p$. We study a Galois cohomology group which is valuable for measuring an obstruction for $K$-rational points on $X$. We analyze a $2$-nilpotent extension of $K$ which contains the information needed for measuring this obstruction. We determine a large subquotient of this Galois cohomology group which arises from Heisenberg extensions of $K$. For $p=3$, we perform a Magma computation with ray class fields, group cohomology, and Galois cohomology, which determin…

Primary rank of the class group of real cyclotomic fields

Let $H= \mathbb{Q}(ζ_{n} + {ζ_{n}}^{-1})$ and $\ell$ be an odd prime such that $q \equiv 1 \pmod \ell$ for some prime factor $q$ of $n$. We get a bound on the $\ell$-rank of the class group of $H$(under some conditions) in terms of the $\ell$-rank of the class group of real quadratic subfield contained in $H$. This is an extension of a recent work of E. Agathocleous (with alternate hypothesis) where she handles $\ell=3$ case. As an application of our main result we relate the $\ell$-rank of real quadratic subfields of $H$.

Binary Hermitian forms over a cyclotomic field

Let z be a primitive fifth root of unity and let F be the cyclotomic field F=Q(z). Let O be the ring of integers. We compute the Voronoi polyhedron of binary Hermitian forms over F and classify GL_2(O)-conjugacy classes of perfect forms. The combinatorial data of this polyhedron can be used to compute the cohomology of the arithmetic group GL_2(O) and Hecke eigen forms.

The Hilbert's class field and the p-class group of the cyclotomic fields

Let $p$ be an irregular prime and $K=\Q(ζ)$ the $p$-cyclotomic field. Let $σ$ be a $\Q$-isomorphism of $K$ generating $Gal(K/\Q)$. Let $S/K$ be a cyclic unramified extension of degree $p$, defined by $S= K(A^{1/p})$ where $A\in K\backslash K^p$, $A\Z_K=\mk a^p$ with $\mk a$ non-principal ideal of $\Z_K$, $A^{σ-μ}\in K^p$ and $μ\in{\bf F}_p$. We compute explicitly the decomposition of the prime $p$ in the subfields $M$ of $S$ of degree $[M:\Q]=p$.

某些实二次域的割圆$\mathbb{Z}_{2}$-扩张的岩泽模

Iwasawa module of the cyclotomic $\mathbb{Z}_{2}$-extension of certain real quadratic fields

本文针对实二次域的割圆$\mathbb{Z}_{2}$-扩张，在岩泽模循环的情形下构造了Greenberg猜想成立的新例子，并利用双二次域的基本单位系给出了计算岩泽模阶数的方法。

全实分圆域上的迭代余切差分积分

Iterated cotangent-difference integrals over totally real cyclotomic fields

本文研究全实分圆域上的迭代余切差分积分，将其表示为分圆多重zeta值，并证明这些积分是混合Tate动机的周期，生成Hopf子代数。

关于某些实二次域的分圆$\mathbb{Z}_p$-扩张上椭圆曲线的模性

On the modularity of elliptic curves over the cyclotomic $\mathbb{Z}_p$-extension of some real quadratic fields

本文遵循Thorne的思想，利用自守性提升定理与Iwasawa理论，在给定假设下证明了某些实二次域的$\mathbb{Z}_p$-扩张上椭圆曲线的模性。

On Weil Numbers in Cyclotomic Fields

In this note, we investigate the p-adic behavior of Weil numbers in the cyclotomic $\mathbb Z\_p$-extension of $\mathbb Q(ζ\_p).$ We determlne the characteristic ideal of the quotient of semi-local units by Weil numbers in terms of the characteristic ideals of some classical modules that appear in Iwasawa Theory.

素数分圆域的相对类数的Kummer比值

The Kummer ratio of the relative class number for prime cyclotomic fields

本文在无Siegel零点、存在Siegel零点及仅对奇特征Dirichlet L-级数假设Riemann Hypothesis三种情形下，显著改进了素数分圆域相对类数Kummer比值的已知界，并通过数值计算验证了理论结果。

Singular integers and p-class group of cyclotomic fields

Let $p$ be an irregular prime. Let $K=\Q(ζ)$ be the $p$-cyclotomic field. From Kummer and class field theory, there exist Galois extensions $S/\Q$ of degree $p(p-1)$ such that $S/K$ is a cyclic unramified extension of degree $[S:K]=p$. We give an algebraic construction of the subfields $M$ of $S$ with degree $[M:\Q]=p$ and an explicit formula for the prime decomposition and ramification of the prime number $p$ in the extensions $S/K$, $M/\Q$ and $S/M$. In the last section, we examine the consequences of these results for the Vandiver's conjecture. This article is at elementary level on Classic…

关于某些分圆 $\mathbb{Z}_2$-扩张的 Hilbert $2$-类域塔

On the Hilbert $2$-class field towers of some cyclotomic $\mathbb{Z}_2$-extensions

本文研究特殊 Dirichlet 域分圆 $\mathbb{Z}_2$-扩张各层的 $2$-类域塔长度、Galois 群结构及 capitulation 问题。

Construction of class fields over cyclotomic fields

Let $\ell$ and $p$ be odd primes. For a positive integer $μ$ let $k_μ$ be the ray class field of $k=\mathbb{Q}(e^{2πi/\ell})$ modulo $2p^μ$. We present certain class fields $K_μ$ of $k$ such that $k_μ\leq K_μ\leq k_{μ+1}$, and find the degree of $K_μ/k_μ$ explicitly. And we also construct, in the sense of Hilbert, primitive generators of the field $K_μ$ over $k_μ$ by using Shimura's reciprocity law and special values of theta constants.

五次和分圆域上的平方和更优结果

Even better sums of squares over quintic and cyclotomic fields

分类了所有次数不超过5且存在有理整数系数万有二次型的全实域，解决了次数≤5的万有型提升问题，主要工具是每个2的倍数都是平方和的域的密集计算分类，并推广到高次实分圆域证明了Kitaoka猜想。

Asymptotic behavior of class groups and cyclotomic Iwasawa theory of elliptic curves

In this article, we study a relation between certain quotients of ideal class groups and the cyclotomic Iwasawa module $X_\infty$ of the Pontrjagin dual of the fine Selmer group of an elliptic curve $E$ defined over $\mathbb{Q}$. We consider the Galois extension field $K^E_n$ of $\mathbb{Q}$ generated by coordinates of all $p^n$-torsion points of $E$, and introduce a quotient $A^E_n$ of the $p$-sylow subgroup of the ideal class group of $K^E_n$ cut out by the modulo $p^n$ Galois representation $E[p^n]$. We describe the asymptotic behavior of $A^E_n$ by using the Iwasawa module $X_\infty$. In p…

On the magnitudes of some small cyclotomic integers

We prove the last of five outstanding conjectures made by R.M. Robinson from 1965 concerning small cyclotomic integers. In particular, given any cyclotomic integer $β$ all of whose conjugates have absolute value at most 5, we prove that the largest such conjugate has absolute value one of four explicit types given by two infinite classes and two exceptional cases. We also extend this result by showing that with the addition of one form, the conjecture is true for $β$ with magnitudes up to 5+1/25.

pi-adic approach of p-class group and unit group of p-cyclotomic fields

Let p > 2 be a prime. Let Q(zeta) be the p-cyclotomic field. Let pi be the prime ideal of Q(zeta) lying over p. This article aims to describe some pi-adic congruences characterizing the structure of the p-class group and of the unit group of the field Q(zeta). For the unit group, this article supplements the 1954 and 1956 papers of Denes on this topic. A complete summarizing of the results obtained follows in the Introduction section of the paper (pages 3 to 6). This new version of the article with the same title, submitted with reference math.NT/0407430 25 Jul 2004: - corrects several typing …

Carlitz module analogues of Mersenne primes, Wieferich primes, and certain prime elements in cyclotomic function fields

In this paper, we introduce a Carlitz module analogue of Mersenne primes, and prove Carlitz module analogues of several classical results concerning Mersenne primes. In contrast to the classical case, we can show that there are infinitely many composite Mersenne numbers. We also study the acquaintances of Mersenne primes including Wieferich and non-Wieferich primes in the Carlitz module context that were first introduced by Dinesh Thakur.

On explicit factors of Cyclotomic polynomials over finite fields

We study the explicit factorization of $2^n r$-th cyclotomic polynomials over finite field $\mathbb{F}_q$ where $q, r$ are odd with $(r, q) =1$. We show that all irreducible factors of $2^n r$-th cyclotomic polynomials can be obtained easily from irreducible factors of cyclotomic polynomials of small orders. In particular, we obtain the explicit factorization of $2^n 5$-th cyclotomic polynomials over finite fields and construct several classes of irreducible polynomials of degree $2^{n-2}$ with fewer than 5 terms. The reciprocals of these irreducible polynomials are irreducible polynomials of …

Cyclotomic Coincidences

In this paper, we show that if $m$ and $n$ are distinct positive integers and $x$ is a nonzero real number with $Φ_m(x)=Φ_n(x)$, then $\frac{1}{2}<|x|<2$ except when $\{m,n\}=\{2,6\}$ and $x=2$. We also observe that 2 appears to be the largest limit point of the set of values of $x$ for which $Φ_m(x)=Φ_n(x)$ for some $m\neq n$.

Infinite products involving Dirichlet characters and cyclotomic polynomials

Using some basic properties of the gamma function, we evaluate a simple class of infinite products involving Dirichlet characters as a finite product of gamma functions and, in the case of odd characters, as a finite product of sines. As a consequence we obtain evaluations of certain multiple $L$-series. In the final part of this paper we derive expressions for infinite products of cyclotomic polynomials, again as finite products of gamma or of sine functions.

Iwasawa's constant $μ$ vanishes in cyclotomic $\Z_p$-extensions of CM fields

Let $\K$ be a galois CM extension of $\Q$ and $\K_{\infty}$ its cyclotomic $\Z_p$-extension. Let $A_n$ be the $p$-parts of the class groups in the intermediate subfields $\K_n \subset \K_{\infty}$ and $\rg{A} = \varprojlim_n A_n$. We show that the $p$-rank of $\rg{A}$ is finite, which is equivalent to the vanishing of Iwasawa's constant $μ$ for $\rg{A}$. (Currently withdrawn)

Ordinary deformations are unobstructed in the cyclotomic limit

The deformation theory of ordinary representations of the absolute Galois groups of totally real number fields (over a finite field $k$) has been studied for a long time, starting with the work of Hida, Mazur and Tilouine, and continued by Wiles and others. Hida has studied the behaviour of these deformations when one considers the $p$-cyclotomic tower of extensions of the field. In the limit, one obtains a deformation ring $R_\infty$ classifying the ordinary deformations of the (Galois group of) the $p$-cyclotomic extension. We show that if $R_\infty$ is Noetherian and certain adjoint $μ$-inv…

Complete classification of torsion of elliptic curves over quadratic cyclotomic fields

In a previous paper, the author examined the possible torsions of an elliptic curve over the quadratic fields $\mathbb Q(i)$ and $\mathbb Q(\sqrt{-3})$. Although all the possible torsions were found if the elliptic curve has rational coefficients, we were unable to eliminate some possibilities for the torsion if the elliptic curve has coefficients that are not rational. In this note, by finding all the points of two hyperelliptic curves over $\mathbb Q(i)$ and $\mathbb Q(\sqrt{-3})$, we solve this problem completely and thus obtain a classification of all possible torsions of elliptic curves o…

分圆伊瓦萨理论中精细类群的局部结构

The local structure of refined class groups in cyclotomic Iwasawa theory

该研究给出分圆伊瓦萨理论中射线类精细结构的显式局部描述，证明了特定条件下 $\mathbb{Q}(\zeta_p)$ 的 $p$ 主射线类群分量非零的等价条件，导出伊瓦萨主猜想的射线类形式，恢复 Finat 公式并推导出精细 $\mu$ 消失结果。

涉及Jacobi和实部及相关分圆矩阵的乘积

Products involving the real parts of Jacobi sums and related cyclotomic matrices

本文研究奇素数幂q下乘积R_q(χ_q)的算术性质，该乘积与Jacobi和的实部相关，并揭示了其与分圆矩阵的联系。

Vanishing of eigenspaces and cyclotomic fields

We use a result of Thaine to give an alternative proof of the fact that, for a prime p>3 congruent to 3 modulo 4, the component e_{(p+1)/2} of the p-Sylow subgroup of the ideal class group of \mathbb Q(ζ_{p}) is trivial.